Monday, 16 July 2012

Eliminating Relata, II

The underlying problem for the ontic structuralism programme is to make mathematical sense of the notion of an "abstract structure", a structure lacking some special domain or carrier set upon which the distinguished relations live. This domain would contain the relata which one wants to eliminate. But the usual notion of a structure, or model, is $\mathcal{A} = (A, R_1, ...)$, where $A$ is the domain or carrier set. The distinguished relations $R_i$ are then subsets of Cartesian products of $A$. In other words, such a model is a "structured set". E.g., the ordering $(\mathbb{N}, <)$ or the field $(\mathbb{R}, 0, 1, +, \times)$. (We think of $\mathbb{N}$ as a set; though it doesn't matter which one it is: usually, it's the finite ordinals $\omega$.) Consequently, if you eliminate the carrier set, then everything else goes with it.

The approach described in the previous post starts with a model (structured set) $\mathcal{A}$, and then identifies the abstract structure with a kind of ramsified proposition that categorically axiomatizes $\mathcal{A}$.

There is another approach. I'd read about this several years ago, but I'd forgotten about it. It's category-theoretic and it was described to me by one of our MCMP graduate students, Hans-Christoph Kotzsch, a couple of weeks ago. On this view, abstract structures are objects in categories. The objects in a category $C$ needn't be regarded as built-up from a carrier set. And one can talk about something akin to "elements" of an object $X$ in a category $C$ by identifying such elements with morphisms $1 \rightarrow X$, where $1$ is a terminal object of $C$.

The idea is developed in a 2011 Synthese article, "Category-Theoretic Structure and Radical Ontic Structural Realism" by Jonathan Bain (and in these slides too).